No need for an air pump.
You'll want to google the aquarium cycle for lots of detail, but in a nutshell, your turtle excretes ammonia, which as it builds up is not healthy-- worse for fish that are breathing it than a turtle. However, there are bacteria that eat that ammonia and excrete it as nitrite. That's even more unhealthy, but there are other bacteria that turn that into nitrate. Nitrate isn't healthy either, but is the least unhealthy of them all. It gets removed during water changes.
These bacteria are primarily growing in your biomedia. That's one of the big reasons for having a filter. However, they are slow growing. It can take weeks for a tank to "cycle," reliably turn ammonia into nitrate. Cloudiness can be a step in that cycle process.
You can get test kits for ammonia, nitrite and nitrate, among other things. I use
http://www.apifishcare.com/product.php? ... d=0&id=580 There are test strips, but they are not as reliable as the drop tests. You can pick up a test kit at any fish store, or get it cheaper online.
When a tank is cycling, testing for all three forms of nitrogen can tell you where you are. Once it is cycled, unless something happens to kill off the beneficial bacteria, you'll only need to test for nitrates to help determine when water changes are necessary.
The general recommendation when a tank is cycling is to avoid water changes until it is done. Doing so can lengthen the cycle. However that advice should be mitigated if the levels of harmful nitrogen (ammonia/nitrite) get very high. With fish, since they breathe the water, you have the potential for death with a high level of either. Since a turtle is breathing air and just swallowing a little of the water, the outcome is not as dire.